Ualbai Umirbaev

The tame and the wild automorphisms of polynomial rings in three variables

Let C = F [x1, x2, . . . , xn] be the polynomial ring in the variables x1, x2, . . . , xn over a field F , and let AutC be the group of automorphisms of C as an algebra over F . An automorphism τ ∈ AutC is called elementary if it has a form τ : (x1, . . . , xi−1, xi, xi+1, . . . , xn) 7→ (x1, . . . , xi−1, αxi + f, xi+1, . . . , xn), where 0 6= α ∈ F, f ∈ F [x1, . . . , xi−1, xi+1, . . . , xn]. The subgroup of AutC generated by all the elementary automorphisms is called the tame subgroup, and the elements from this subgroup are called tame automorphisms of C. Non-tame automorphisms of the algebra C are called wild. It is well known [6], [9], [10], [11] that the automorphisms of polynomial rings and free associative algebras in two variables are tame. At present, a few new proofs of these results have been found (see [5], [8]). However, in the case of three or more variables the similar question was open and known as “The generation gap problem” [2], [3] or “Tame generators problem” [8]. The general belief was that the answer is negative, and there were several candidate counterexamples (see [5], [8], [12], [7], [19]). The best known of them is the following automorphism σ ∈ Aut(F [x, y, z]), constructed by Nagata in 1972 (see [12]): σ(x) = x+ (x − yz)z, σ(y) = y + 2(x − yz)x+ (x − yz)z, σ(z) = z.

Poisson brackets and two-generated subalgebras of rings of polynomials

Let A = F [x1, x2, . . . , xn] be a ring of polynomials over a field F on the variables x1, x2, . . . , xn. It is well known (see, for example, [11]) that the study of automorphisms of the algebra A is closely related with the description of its subalgebras. By the theorem of P. M. Cohn [4], a subalgebra of the algebra F [x] is free if and only if it is integrally closed. The theorem of A. Zaks [13] says that the Dedekind subalgebras of the algebra A are rings of polynomials in a single variable. A. Nowicki and M. Nagata [8] proved that the kernel of any nontrivial derivation of the algebra F [x, y], char(F ) = 0, is also a ring of polynomials in a single generator. An original solution of the occurrence problem for the algebra A, using the Groebner basis, was given by D. Shannon and M. Sweedler [9]. However, the method of the Groebner basis does not give any information about the structure of concrete subalgebras. Recall that the solubility of the occurrence problem for rings of polynomials over fields of characteristic 0 was proved earlier by G. Noskov [7]. The present paper is devoted to the investigation of the structure of twogenerated subalgebras of A. In the sequel, we always assume that F is an arbitrary field of characteristic 0. Let us denote by f the highest homogeneous part of an element f ∈ A, and by 〈f1, f2, . . . , fk〉 the subalgebra of A generated by the elements f1, f2, . . . , fk ∈ A. Definition 1. A pair of polynomials f1, f2 ∈ A is called ∗-reduced if they satisfy the following conditions: 1) f1, f2 are algebraically dependent; 2) f1, f2 are algebraically independent; 3) f1 / ∈ 〈f2〉, f2 / ∈ 〈f1〉. Recall that a pair f1, f2 with condition 3) is usually called reduced. Condition 1) means that we exclude the trivial case when f1, f2 are algebraically independent. We do not consider the case when f1, f2 are algebraically dependent. Concerning this case, recall the well-known theorem of S. S. Abhyankar and T. -T. Moh [1], which says that if f, g ∈ F [x] and 〈f, g〉 = F [x], then f ∈ 〈ḡ〉 or ḡ ∈ 〈f〉.

The Nagata automorphism is wild

It is proved that the well known Nagata automorphism of the polynomial ring in three variables over a field of characteristic zero is wild, that is, it can not be decomposed into a product of elementary automorphisms.

Generic, almost primitive and test elements of free Lie algebras

We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.

THE FREIHEITSSATZ AND THE AUTOMORPHISMS OF FREE RIGHT-SYMMETRIC ALGEBRAS

We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame.

Centralizers in free Poisson algebras

We prove an analog of the Bergman Centralizer Theorem for free Poisson algebras over an arbitrary field of characteristic 0. Some open problems are formulated.

Subalgebras of free leibniz algebras

Leibniz algebras are in some sense non-(anti)commutative analogs of Lie algebras. The variety of all Lie algebras has the Schreier property (i.e. any subalgebra of the free algebra is free). This is not the case in the variety of all Leibniz algebras. Nevertheless we prove that: 1) the variety of all Leibniz algebras has the property of differential separability for subalgebras; 2) the Jacobian conjecture is true for free Leibniz algebras; 3) the free Leibniz algebras are finitely separable (in particular, it follows that the occurrence problem for free Leibniz algebras is solvable).

Defining relations of the tame automorphism group of polynomial algebras in three variables

Abstract We describe a set of defining relations of the tame automorphism group TA 3(F) of the polynomial algebra F [x1, x2, x3] in variables x1, x2, x3 over an arbitrary field F of characteristic 0.

AUTOMORPHISMS OF TWO-GENERATED FREE LEIBNIZ ALGEBRAS

We obtain a characterization of tame automorphisms of the free Leibniz algebra in two variables. It gives an algorithm to recognize tame automorphisms. Using these results we construct a wild automorphism of this algebra. *Partially supported by CRCG Research Grant 2550/301/01.

The Magnus embedding for right-symmetric algebras

We construct a basis for the universal multiplicative enveloping algebra U(A) of a right-symmetric algebra A. We prove an analog of the Magnus embedding for right-symmetric algebras; i.e., we prove that a right-symmetric algebra A/R2, where A is a free right-symmetric algebra, is embedded into the algebra of triangular matrices of the second order.